Odd-order cyclic group is characteristic in holomorph

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This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., odd-order cyclic group) must also satisfy the second group property (i.e., holomorph-characteristic group)
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Statement

Any Odd-order cyclic group (?) is a Characteristic subgroup (?) inside its holomorph.

Related facts

Related facts about odd-order cyclic groups

Breakdown for other cyclic groups

Other related facts

Facts used

  1. Odd-order cyclic group equals commutator subgroup of holomorph
  2. Commutator subgroup is characteristic

Proof

The proof follows directly from facts (1) and (2).